Let \(\tilde{\ell }\)  be a word length function on \(H\) . Since \(H\)  is simply connected nilpotent,
\(\log (1+|\eta |) \simeq \log (1+ \tilde{\ell }(\exp (\eta ))\quad \text{on~$\mathfrak {h}$}\) 
(this fact is well known; see, e.g., [1]}). The rest is clear.
